Distribution of Exceedances
Since Peaks over Threshold (POT) focuses on the realizations exceeding a given (high)
threshold and the threshold method uses data more efficiently, we concentrate on peak over
threshold approach.
The Pickands-Balkema-de Haan Theorem
Suppose that X 1 , X 2 ,..., X n are n independent realisations of a random variable X with a
distribution function F ( x ) . Let xF be the finite or infinite right endpoint of the distribution
F. The distribution function of the excesses over certain (high) threshold u is given by
Fu ( x ) = Pr { X − u ≤ x X > u} =
F ( x + u ) − F (u )
1 − F (u )
for 0 ≤ x < xF − u.
The Pickands-Balkema-de Haan theorem states that if the distribution function F ∈ DA ( H ξ )
then there exists a positive measurable function σ (u ) such that
lim
u→x F
sup { F ( x) − Gξ σ
, (u )
u
}
( x) = 0
0 ≤ x < xF − u
and vice versa, where Gξ ,σ (u ) ( x) denotes the Generalised Pareto distribution (see below).
The above theorem states that as the threshold u becomes large, the distribution of the
excesses over the threshold tends to the Generalised Pareto distribution, provided the
underlying distribution F belongs to the domain of attraction of the Generalised Extreme
Value distribution.
Generalized Pareto Distribution (GPD)
The GPD is a two parameter distribution with distribution function
1
−1
ξ
ξx
1
1
−
+
β
Gξ , β ( x ) =
1 − exp − x
β
(
)
(
)
where β > 0, and where x ≥ 0 when ξ ≥ 0 and 0 ≤ x ≤ − β
ξ ≠0
ξ =0
ξ when ξ 0 then ξ , β is a reparametrized version of the ordinary
Pareto distribution, which has a long history in actuarial mathematics as a model for large
losses; ξ = 0 corresponds to the exponential distribution and ξ < 0 is known as a Pareto type
II distribution.
The first case is the most relevant for risk management purposes since the GPD is heavytailed when ξ > 0. Whereas the normal distribution has moments...